# Systematic Approach for the Test Data Generation and Validation of ISC/ESC Detection Methods

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Reduction in heat influx from adjacent TRs to slow down or rather prevent TP in accordance with the US Vehicle Battery Safety Roadmap Guidance, which states that TP must not be initiated [17] if a never fully excluded cell-level TR occurs [18]. Prevention by smart module design [19,20], active or passive cooling strategies [21,22] and/or thermal isolation [23].
- Detection of battery faults and abnormal conditions for countermeasures, warning and evacuation before a hazardous situation develops. Here, the Global Technical Regulation on Electrical Vehicle Safety (GTR-EVS) specifies at least 5 $\mathrm{min}$ or enough time for egress [24].

- Extensive literature review of disturbances on the measurement signal and their magnitudes;
- Summary of common qualitative and quantitative evaluation criteria;
- Generation of test data with stochastic disturbances and variations with consideration of both fault-free and fault-containing samples with the scope of ISC and ESC;
- Example comparison based on binary classifiers and identification of optimum parameter combinations.

## 2. State of the Art

- Complexity or difficulty of the application, e.g.,
- -
- -
- Large fault model parameter sets [30];
- -
- -
- -
- -

- Simplifications and assumptions concerning:

#### 2.1. Measurement Uncertainty

#### 2.2. Cell-to-Cell Variations

- Orientation at statistical founded experimentally determined variations;
- Assessment of the worst case boundaries.

**Table 4.**Assumptions of CtCV for both capacity (C) and resistance (R) utilized in recent studies in the context of battery fault detection evaluation. For three studies, no cell type was specified. Please refer to Table 3 for comparison with experimental determined CtCV values.

Author et al. | Year | Cell | ${\mathit{C}}_{\mathbf{\text{nom.}}}$/Ah | $\mathbf{\Delta}\mathit{R}/\%$ | $\mathbf{\Delta}\mathit{C}/\%$ | Source |
---|---|---|---|---|---|---|

Dey | 2016 | 5, 10 and 15 | [77] | |||

Chang | 2019 | 18650 cell | 2 | 10, 20 and 40 | 20 | [130] |

Chen | 2019 | A123 ANR26650-M1A | 2300 | ±3 | [30] | |

Dubarry | 2019 | 0.0, 3.75, 7.5, 12.5 and 15 | 0.0, 1.25, 2.5, 3.75 and 5 | [131] | ||

Zhang | 2019 | −5, −3, 2 and 5 | −5, −3, 2 and 5 | [68] | ||

Schmid | 2021 | Samsung INR18650-25R | 2500 | +10 | [42] | |

Song | 2021 | 60 | 0, 1.5 and 2.8 | [132] |

#### Voltage Offset

^{−1}to 0.2 mV K

^{−1}[135], no general statement of the effect can be made. With respect to published maximum temperature differences inside battery modules of <10 K [136,137,138,139,140] the voltage variation is expected to be <1 mV. In addition, the already mentioned CtCV causes further voltage variations since the differences in internal resistance will cause slight variations of the voltage-drop and overvoltage during charge and discharge, respectively.

#### 2.3. Evaluation Aspects

${t}_{\mathrm{p}}$ | True positive | ${t}_{\mathrm{n}}$ | True negative | |

${f}_{\mathrm{p}}$ | False positive | ${f}_{\mathrm{n}}$ | False negative |

## 3. Material and Methods

#### 3.1. Reference Cell

#### 3.2. Model

^{®}[152] with pre- and post-processing was performed in native Matlab. As displayed in Figure 2, a second order ECM was chosen, which is in accordance with many other studies, where either a first or second order model was chosen as compromise between accuracy and complexity as investigated by Zhang et al. [153].

- Parameterization is doable by standard electrochemical tests;
- Implementation of parameter distribution is simplified;
- Fault representation (see below) is well-defined;
- Simulation time is fast.

#### 3.2.1. ISC/ESC-Fault Representation

#### 3.2.2. Randomness and Variation

#### Measurement Uncertainty

#### Cell-to-Cell Variation

#### 3.2.3. Parameterization

#### 3.3. Simulation Cases

- The fault chance is 80%;
- Only one cell fault per time;
- Only one fault event per simulation run.

#### 3.4. Fault Detection Methods

- Generate many samples without presence of a fault.
- Calculate the fault signals for the detection method for each sample.
- Determine the maximal fault signal value for each sample.
- Define the threshold $\zeta $ as $\zeta =\mu +\lambda \sigma $.
- If the fault signal is greater than $\zeta $ a fault will be assumed.

#### 3.4.1. Deviation from Mean

#### 3.4.2. z-Score

## 4. Results and Discussion

#### 4.1. Number of Simulations

#### 4.2. Distribution of Fault Feature

#### 4.3. Fault Detection

#### 4.4. Further Investigations

#### 4.4.1. Threshold Level

#### 4.4.2. Noise Level

#### 4.4.3. CtCV

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BMS | Battery management system |

CC | Constant current |

CtCV | Cell-to-cell variation |

CV | Coefficient of variation |

ECM | Equivalent circuit model |

ESC | External short circuit |

EV | Electric vehicle |

FNR | False negative rate (specificity) |

FPR | False positive rate |

GTR | Global Technical Regulation |

GTR-EVS | Global Technical Regulation on Electrical Vehicle Safety |

GUM | Guide to the expression of uncertainty in measurement |

IC | Integrated circuit |

ISC | Internal short circuit |

LIB | Lithium-ion battery |

LUT | Look-up-table |

MA | Moving average |

NPV | Negative predictive value |

NRMSE | Normalized root mean squared error |

OCV | Open circuit voltage |

P2D | Pseudo two-dimensional |

P-OCV | Pseudo open circuit voltage |

PPV | Positive predictive value |

RMS | Root mean square |

RMSE | Root mean squared error |

ROC | Receiver operating characteristic |

SOC | State of Charge |

SVM | Support vector machine |

TNR | True negative rate |

TPR | True positive rate (sensitivity) |

TR | Thermal runaway |

WLTP | Worldwide Harmonized Light-Duty Vehicles Test Procedure |

Y | Youden-Index |

## Appendix A

#### Appendix A.1. Evaluation of Computational Effort

Listing 1. Implementation of the moving average algorithms using functions from pandas, NumPy and Numba. |

import NumPy as np |

from Numba import njit, prange, float64, int16 |

def rollingMeanPandas(data, w=10): |

return data.rolling(w).mean() |

def rollingMeanNumPy(data, w=10): |

result=np.empty_like(data) |

for row in range(data.shape[0]): |

window=np.zeros((w, data.shape[1])) |

window[:]=np.nan # Initialise with np.nan |

# Relevant for the first w rows |

tmp=data[max(0,row−w+1):row+1, :] # Selection of data with window w |

window[−len(tmp):, :]=tmp |

result[row]= np.mean(window,axis=0) # Calculate mean over each column selection |

return result |

@njit(float64[:,:](float64[:,:],int16), parallel = True) # See above rollingMeanNumPy |

def rollingMeanNumba(data, w=10): |

result=np.empty_like(data) |

for row in prange(data.shape[0]): |

window=np.zeros((w, data.shape[1])) |

window[:]=np.nan |

tmp=data[max(0,row−w+1):row+1, :] |

window[−len(tmp):, :]=tmp |

avg=np.empty(window.shape[1], dtype=float64) |

# np.mean(axis=0) is not implemented by Numba−>custom calculation |

for col in range(window.shape[1]): |

avg[col]=window[:,col].mean() |

result[row]=avg |

return result |

Listing 2. Import of both functions and required packages. Random generation of test data with two different dimensions. |

from SampleFunctions import ∗ |

import pandas as pd |

import NumPy as np |

sampleData=np.random.rand(100000,12) |

# SampleData=np.random.rand(100000,100) |

sampleDF=pd.DataFrame(sampleData) |

Listing 3. Evaluation of the computational time for each implemented function with respect to the required data structure. |

timeit rollingMeanPandas(sampleDF, 10) |

timeit rollingMeanNumba(sampleData, 10) |

timeit rollingMeanNumPy(sampleData, 10) |

Listing 4. Validation of correct implementation by pair-to-pair comparison of the calculated results based on the same random test data. |

# Comparison of the evaluated arrays |

print(np.allclose(rollingMeanNumba(sampleData, 10), |

rollingMeanPandas(sampleDF, 10), equal_nan=True)) |

print(np.allclose(rollingMeanNumPy(sampleData, 10), |

rollingMeanPandas(sampleDF, 10), equal_nan=True)) |

print(np.allclose(rollingMeanNumba(sampleData, 10), |

rollingMeanNumPy(sampleData, 10), equal_nan=True)) |

**Table A1.**Technical specifications utilized to calculate the moving average on both a standard notebook (A) and a simulation workstation (B).

Specification | A | B |
---|---|---|

Processor | Intel Core i5-8265U | Intel Xeon W-2275 |

Total cores | 4 | 14 |

RAM | 8 GB | 256 GB |

Year | 2020 | 2022 |

**Table A2.**Computational times of the investigated moving average implementations on both standard notebook (A) and simulation workstation (B) and sample sizes.

A | B | |||
---|---|---|---|---|

Implementation | $\mathit{n}=12$ | $\mathit{n}=100$ | $\mathit{n}=12$ | $\mathit{n}=100$ |

Pandas | 114 ms | 63.7 ms | 41.3 ms | 573 ms |

NumPy | 2.34 s | 1.93 s | 1.34 s | 1.56 s |

Numba | 23.1 ms | 18.3 ms | 15.2 ms | 24.1 ms |

#### Appendix A.2. Consistency of Separate Simulation Studies

**Table A3.**Achieved detection quality of z-score method with threshold level of $\lambda =3$ for repetitive simulation of the default simulation case with no-fault condition (left) and with 80% failure rate (right). Results were obtained on the basis of 1200 and 2400 repetitions for fault-free and fault datasets, respectively. The mean $\mu $ of the maximum fault signal per simulation run is also given. For detailed information on the given indicators FPR and TNR please refer to Table 6.

$\mathit{\mu}$ | FPR/% | TNR/% | FPR/% | TNR/% | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

No | I | II | I | II | I | II | No | I | II | I | II |

$\mathit{w}$ | $\mathit{w}$ | ||||||||||

1 | 3.108 | 3.110 | 0.167 | 0.167 | 99.833 | 99.833 | 1 | 0.600 | 0.832 | 99.400 | 99.168 |

10 | 1.369 | 1.364 | 1.000 | 0.833 | 99.000 | 99.167 | 10 | 2.183 | 3.854 | 97.817 | 96.146 |

100 | 0.408 | 0.408 | 1.083 | 1.500 | 98.917 | 98.500 | 100 | 3.523 | 2.474 | 96.477 | 97.526 |

1000 | 0.111 | 0.111 | 1.083 | 0.917 | 98.917 | 99.083 | 1000 | 1.394 | 2.053 | 98.606 | 97.947 |

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**Figure 1.**Workflow for generating a dataset with variable characteristics (disturbances and faults) for setting up and validating different fault detection methods. External inputs represent parameter presets that are used either in the Monte Carlo-like data generation process or for different fault detection configurations.

**Figure 2.**Second order ECM as implemented in this simulation study to emulate the dynamic behaviour of one cell. All parameters describing the normal operation of the cell are implemented dependent on the SOC. Parallel simulation of multiple models results in the dynamic characteristics of one module in ks1p configuration. Emulation of ISC-fault by parallel resistance is indicated in red.

**Figure 3.**Simulated voltages for faulty cell (${C}_{11}$) and fault-free cell (here ${C}_{01}$) for simulation of 1 $\Omega $ ISC-fault at 1518 s for 85 s. The period of fault is magnified at the right and marked in all axis in red colour. The corresponding z-score fault signal with $w=10$ ( ${f}_{z}^{10}$) is given in the lower figure, as well as the $3\sigma $ threshold level.

**Figure 4.**

**Left**: Classification of simulation runs to ${\mathrm{true}}_{\mathrm{positive}}$, ${\mathrm{false}}_{\mathrm{positive}}$ and ${\mathrm{false}}_{\mathrm{negative}}$ with respect to the fault resistance ${R}_{\mathrm{ISC}}$ and fault duration $\Delta {t}_{\mathrm{ISC}}$ for z-score and window size $w=10$. Please note that ${\mathrm{true}}_{\mathrm{negative}}$ (see Section 2.3) will not appear in this representation. The boundary between ${t}_{\mathrm{p}}$ and ${f}_{\mathrm{n}}$ is visualized by fitted model using linear support vector classification (SVC).

**Right**: Decision boundaries for both detection methods and variable window sizes.

**Figure 5.**Achieved detection times $\Delta {t}_{\mathrm{detection}}$ of the z-score method ($\lambda =3,w=10$) with respect to (

**a**) the fault resistance ${R}_{\mathrm{ISC}}$, (

**b**) the fault duration $\Delta {t}_{\mathrm{ISC}}$ and (

**c**) the time of fault ${t}_{\mathrm{ISC}}$. Please note that only ${t}_{\mathrm{p}}$ classified cases are considered in this analysis.

**Figure 6.**Achieved detection quality for both methods $\Delta \mu $ and z-score with respect to the underlying threshold level $\lambda $ and filter size w;

**Left**: Youden-index (Y).

**Right**: Approximation of criticality of faults that were not detected ($\kappa $).

**Figure 7.**Achieved detection quality for both methods $\Delta \mu $ and z-score with respect to the underlying filter size w dependent on the threshold level $\lambda $ and corresponding threshold $\zeta $ expressed by the Youden-index (Y).

**Figure 8.**Achieved classification accuracy of both methods $\Delta \mu $ and z-score (hatch) at discrete window sizes w under the influence of various levels of measurement noise $\Delta U\sim \mathcal{N}(0,{\sigma}_{U})$. The result corresponding to each threshold level $\lambda $ is indicated by the alpha level.

**Figure 9.**Achieved classification accuracy of both methods $\Delta \mu $ and z-score (hatch) at discrete window sizes w under the influence of various kinds of disturbances. In addition to the default case with $\Delta U$, parameter variation $\Delta Z$ and $\Delta \mathrm{OCV}$ as well as the combination of them was added. The result corresponding to each threshold level is indicated by the alpha level.

**Table 1.**Assumptions for the level of measurement uncertainty for the common battery system quantities cell voltage (U), current (I) and temperature ($\vartheta $) if modelled by zero-mean Gaussian noise with standard deviation $\sigma $. Displayed values were derived from publication if standard deviation was not given. Please refer to the table footnotes for limitations due to the provided data.

Author et al. | ${\mathit{\sigma}}_{\mathit{U}}/\mathbf{mV}$ | ${\mathit{\sigma}}_{\mathit{I}}/\mathbf{mA}$ | ${\mathit{\sigma}}_{\mathit{\vartheta}}/\xb0\mathbf{C}$ | Source |
---|---|---|---|---|

Alavi | 0.316 | [89] * | ||

Dey | 50 | 0.08 | 0.5 | [55] |

Dey | 100 | 3.16 | 0.447 | [56] * |

Dey | 5 | 10 | 0.3 | [90] |

Dey | 5 | 10 | 0.3 | [91] |

Feng | 2 | 0.1 | [88] | |

Feng | 1 | 0.01 | [88] *^{,1} | |

Kang | 100 | [53] *^{,2} | ||

Kang | 100 | [57] *^{,2} | ||

Kim | 10 | [59] | ||

Pan | 10 | [92] *^{,2} | ||

Shang | 10 | [39] * | ||

Son | 450 | [71] | ||

Xia | 1 | [50] | ||

Zhang | 2 | 10 | [87] * | |

Zhang | 2 | 25 | 0.05 | [93] |

Zhao | 6 | [83] |

^{1}Definition by accuracy.

^{2}Definition by amplitude.

**Table 2.**Reference values describing the measurement uncertainty from real application for common battery system quantities. For better comparability in case of percentages given, the absolute values were calculated based on 3.7 V and 44.4 V as nominal voltages for cell and module levels, respectively. The values derived as such are indicated by parenthesis.

Description | Value | Comment | Source | |
---|---|---|---|---|

Accuracy from analysed SMC-EV ^{1} platform | <10 mV | [94] | ||

Accuracy from investigated EV | ±5 mV with resolution 1 mV | Cell voltage | [45] | |

±1 °C | Cell temperature | [45] | ||

±0.1 A if $I<$30 A else ±1% | Pack current | [45] | ||

±1% | (±444 mV) | Pack voltage | [45] | |

BMS accuracy of EV | ±0.1% | (±37 mV) | General assumption, no source | [57] |

Standard deviation of investigated module | 0.3806 mV | Data from previous study; not published | [14] | |

Accuracy from BMS-IC ^{2} | ±2.8 mV | Cell voltage, max. Value | [95] | |

±2.5% | ( ±1110 mV) | Pack voltage | [95] | |

±5 °C | Temperature | [95] | ||

Accuracy from BMS-IC ^{2} | ±1.4 mV | Cell voltage | [96] |

^{1}Service and Management Center for Electric Vehicles in Beijing.

^{2}Integrated circuit.

**Table 3.**Overview of the literature on experimental determined CtCV of cell capacity and resistance, given as coefficient of variation ($\mathrm{CV}$); see Equation (4). Please refer to Table 2 for comparison with common approximations for CtCV simulation. Cell specifications were taken from source; please refer to Wildfeuer et al. [103] for an in depth analysis of recent studies.

Author et al. | Year | N | Cell | State | ${\mathit{C}}_{\mathbf{nom}}$/Ah | ${\mathbf{CV}}_{\mathit{R}}$/% | ${\mathbf{CV}}_{\mathit{C}}$/% | Source |
---|---|---|---|---|---|---|---|---|

Dubarry | 2009 | 100 | - | - | 0.30 | - | 1.86 | [104] |

2010 | 100 | - | - | 0.30 | 30.12 | 1.86 | [105] | |

2011 | 10 | - | - | 1.90 | 5.66 | 0.16 | [100] | |

Shin | 2013 | 10,000 | - | Model | - | 4.40 | 0.00 | [106] |

Paul | 2013 | 20,000 | - | - | 4.40 | - | 1.30 | [107] |

Zheng | 2013 | 96 | - | - | 70.00 | 19.47 | - | [45] |

Baumhofer | 2014 | 48 | Sanyo/Panasonic UR18650E | - | 1.85 | - | 0.50 | [108] |

Rothgang | 2014 | 700 | HP prismatic Cell | New | - | 2.87 | 2.36 | [109] |

Schuster | 2015 | 954 | IHR18650A | Aged, from EV 2 | 1.95 | 3.19 | 1.57 | [110] |

2015 | 954 | IHR18650A | Aged, from EV 1 | 1.95 | 2.56 | 2.25 | [110] | |

2015 | 484 | IHR18650A | New | 1.95 | 1.94 | 0.80 | [110] | |

Devie | 2016 | 100 | NCR 18650B | New | 3.35 | 0.30 | 0.80 | [111] |

Campestrini | 2016 | 250 | Panasonic NCR18650PD | New | 2.80 | 0.72 | 0.16 | [112] |

An | 2016 | 198 | - | - | 5.30 | 2.85 | 1.34 | [113] |

2016 | 7739 | - | - | 5.30 | - | 1.45 | [114] | |

Rumpf | 2017 | 600 | Sony US26650FTC1 | New, Batch 1 | 3.00 | 1.81 | 0.23 | [102] |

2017 | 1100 | Sony US26650FTC1 | - | 3.00 | - | - | [102] | |

Barreras | 2017 | 208 | SLPB 120216216 | New | 53.00 | 5.63 | 0.35 | [115] |

Rumpf | 2017 | 500 | Sony US26650FTC1 | New, Batch 2 | 3.00 | 0.73 | 0.33 | [102] |

Devie | 2018 | 51 | LG ICR18650 C2 | New | 2.80 | 3.55 | 2.00 | [116] |

2018 | 15 | LG ICR18650 C2 | Aged, 1000 cycles | 2.80 | 5.00 | 2.80 | [116] | |

Oeser | 2018 | 50 | ICR 18650 26F | Aged, 1464 cycles, 77。8% SOH | 2.60 | - | 1.10 | [117] |

Baumann | 2018 | 185 | BatteryPack, GS Yuasa (LEV50) | Aged, from EV | 50.00 | 4.40 | 0.85 | [118] |

2018 | 164 | Panasonic NCR18650PF | Aged, 3 years | 2.90 | 0.92 | 0.35 | [118] | |

Zou | 2018 | 248 | - | New | 3.00 | 0.95 | 0.37 | [119] |

Zilberman | 2019 | 13 | LG Chem INR18650-MJ1 | New | 3.50 | 1.08 | 0.22 | [120] |

2019 | 48 | LG MJ1 | New | 3.35 | 0.68 | 0.20 | [121] | |

2019 | 24 | LG MJ1 | Aged, 10 months | 3.35 | 0.75 | 0.38 | [121] | |

2020 | 48 | LG Chem INR18650-MJ1 | New | 3.35 | 0.79 | 0.20 | [122] | |

Wildfeuer | 2021 | 568 | Sony US18650VTC5A | New | 2.50 | 0.86 | 0.24 | [103] |

Schindler | 2021 | 48 | LG MJ1 | New, Batch 1 | 3.35 | 0.65 | 0.20 | [123] |

2021 | 200 | LG MJ1 | New, Batch 3 | 3.35 | 3.40 | 0.40 | [123] | |

2021 | 160 | LG MJ1 | New, Batch 2 | 3.35 | 1.04 | 0.36 | [123] | |

Oeser | 2022 | 137 | ICR18650-26J | Aged, 2 years | 2.60 | 2.00 | 0.26 | [124] |

2022 | 480 | ICR18650-26J | New | 2.60 | 1.69 | 0.26 | [124] | |

Reiter | 2023 | 14 | - | - | 128.00 | 2.20 | 0.39 | [125] |

Hein | 2023 | 200 | ICR 18650-26J | - | 2.60 | 1.59 | 0.23 | [126] |

**Table 5.**Published values for the balancing hysteresis $\Delta \mathrm{OCV}$ taken from sources close to field-application, such as application guidelines from BMS-manufacturers or accuracy values given for BMS in the academic literature.

Description | $\mathbf{\Delta}$OCV/mV | Comment | Source |
---|---|---|---|

Guideline | 100 | Trigger for balancing | [141] |

Guideline | 10 | Recommendation for ${U}_{\text{max.}}-{U}_{\text{min.}}$ | [142] |

Guideline | 50 | Acceptable static voltage | [143] |

100 | Acceptable dynamic voltage | ||

Application | 20 | Optimized balancing | [144] |

Application | 100 | Common hysteresis | [128] |

Application | 20 | Measurement of EV | [145] |

7 | Experimental balancing |

Symbol | Name | Definition | Used in |
---|---|---|---|

TPR | True positive rate ^{1} | $\frac{{T}_{\mathrm{p}}}{{T}_{\mathrm{p}}+{F}_{\mathrm{n}}}$ | [40] |

FNR | False negative rate ^{2} | $\frac{{F}_{\mathrm{n}}}{{T}_{\mathrm{p}}+{F}_{\mathrm{n}}}$ | [40,41,63,73,79] |

TNR | True negative rate | $\frac{{T}_{\mathrm{n}}}{{T}_{\mathrm{n}}+{F}_{\mathrm{p}}}$ | |

FPR | False positive rate | $\frac{{F}_{\mathrm{p}}}{{T}_{\mathrm{n}}+{F}_{\mathrm{p}}}$ | [41,51,55,63,73,79] |

PPV | Positive predictive value | $\frac{{T}_{\mathrm{p}}}{{T}_{\mathrm{p}}+{F}_{\mathrm{p}}}$ | |

NPV | Negative predictive value | $\frac{{T}_{\mathrm{n}}}{{T}_{\mathrm{n}}+{F}_{\mathrm{n}}}$ | |

Y | Youden-index | TPR + FNR − 1 |

^{1}Alias: Sensitivity;

^{2}Alias: Specificity.

**Table 7.**Selected datasheet properties of the SLPB98106100 pouch cell from Kokam that was used as reference cell for the simulation.

Parameter | Symbol | Value |
---|---|---|

Nominal capacity | ${C}_{\text{nom.}}$ | 10 Ah |

Nominal voltage | ${U}_{\text{nom.}}$ | 3.7 V |

Upper voltage limit | ${U}_{\text{max.}}$ | 4.2 V |

Lower voltage limit | ${U}_{\text{min.}}$ | 2.7 V |

Charge current | ${I}_{\text{nom.}}$|${I}_{\text{max.}}$ | 5 A|20 A |

Discharge current | ${I}_{\text{nom.}}$|${I}_{\text{max.}}$|${I}_{<10s}$ | 5 A|20 A|30 A |

Weight | m | 0.210 kg |

**Table 8.**Parameters of the Monte Carlo data generation, including simulated uncertainty and ISC-fault replication. The individual parameter-set was generated randomly based on either a uniform ($\mathcal{U}$) or a Gaussian ($\mathcal{N}$) distribution. Left: Values for the implemented model disturbances dependent on the simulation case, where measurement uncertainty only is considered as Default. Please refer to Section 2.1 and Section 2.2 for further details on the implementation. Right: Intervals for generation of a fault-simulation parameter-set based on a uniform distribution.

$\mathbf{\Delta}\mathit{U}$ | $\mathbf{\Delta}\mathbf{OCV}$ | $\mathbf{\Delta}\mathit{Z}$ | Range | |||
---|---|---|---|---|---|---|

Distribution | $\sim \mathcal{N}(0,{\mathit{\sigma}}_{\mathit{U}})$ | $\sim \mathcal{U}(-\frac{\mathit{d}}{2},\frac{\mathit{d}}{2})$ | $\sim \mathcal{N}(0,{\mathit{\sigma}}_{\mathit{Z}})$ | Distribution | $\sim \mathcal{U}\left(\mathbf{Range}\right)$ | |

Case | ${\mathit{\sigma}}_{\mathit{U}}$/mV | d/mV | ${\mathit{\sigma}}_{\mathit{Z}}$/% | Parameter | Symbol | |

Default ($\Delta U$) | 1.0 | 0.0 | 0.0 | Cell index of fault | k | $\in [1;N]$ * |

Modified Default | 0.5, 1, 2 and 10 | 0.0 | 0.0 | Time of fault | ${t}_{\mathrm{ISC}}$ | $\in [1;T]\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ ** |

$\Delta U+\Delta \mathrm{OCV}$ or $+\Delta Z$ | 1.0 | 10 | 1.0 | Fault duration | $\Delta {t}_{\mathrm{ISC}}$ | $\in [1;120]\phantom{\rule{3.33333pt}{0ex}}\mathrm{s}$ |

$\Delta U+\Delta \mathrm{OCV}$ and $+\Delta Z$ | 1.0 | 10 | 0.1 | Fault resistance | ${R}_{\mathrm{ISC}}$ | $\in [1;100]\phantom{\rule{3.33333pt}{0ex}}\Omega $ |

**Table 9.**Probability of samples within multiple standard deviations around the mean of a normal distribution. The two-sided values describe $P(\mu -\lambda \sigma \le x\le \mu +\lambda \sigma )$ and for the one-sided case $P(x\le \mu +\lambda \sigma )$. Here, the left side of the distribution is already fully incorporated.

$\mathit{\lambda}$ | 2-Side/% | 1-Side/% |
---|---|---|

1 | 68.27 | 84.13 |

2 | 95.45 | 97.72 |

3 | 99.73 | 99.87 |

**Table 10.**Summary statistic coefficient of variation ($\mathrm{CV}$) for the default simulation case with 300 simulation runs. Evaluated maximum fault signal for deviation from mean $\Delta \mu $ and z-score z dependent on the filter window size w. Required minimal simulation runs N to achieve 2% accuracy results with 95% confidence.

Evaluation | $\mathbf{CV}$/% | ${\mathit{N}}_{\mathbf{\text{min.}}}^{95\%}$ | ||
---|---|---|---|---|

$\mathit{w}$ | $\mathbf{\Delta}\mathit{\mu}$ | $\mathit{z}$ | $\mathbf{\Delta}\mathit{\mu}$ | $\mathit{z}$ |

1 | 5.84 | 1.56 | 33 | 3 |

2 | 5.45 | 3.28 | 29 | 11 |

5 | 6.24 | 5.24 | 38 | 27 |

10 | 5.70 | 5.37 | 32 | 28 |

20 | 6.48 | 6.29 | 41 | 38 |

100 | 8.76 | 8.02 | 74 | 62 |

200 | 8.31 | 7.85 | 67 | 60 |

1000 | 11.17 | 10.80 | 120 | 113 |

**Table 11.**Statistical properties average $\mu $, standard deviation $\sigma $ and skewness ${\mu}_{3}$ for the maximum fault signal distribution of the fault-free simulation setup with $N=1200$. Evaluated of the fault signals for the detection methods z and $\Delta \mu $ for different window sizes w. The corresponding FPR in % is calculated based on a threshold $\zeta $ associated with $3\sigma $ which should result in a FPR of 0,18% according to Table 9. Left margin: Exemplary barplot of the maximum fault signal distribution for the z-score of $w=10$ (green, highlighted values) and approximation by a normal distribution based on the statistical properties $\mu $, $\sigma $ (blue). Skewness is visualized by marked peak position (white).

$\mathit{\mu}$ | $\mathit{\sigma}$ | ${\mathit{\mu}}_{3}$ | FPR/% | ||||||

Evaluation | $\mathbf{\Delta}\mathit{\mu}$ | z | $\mathbf{\Delta}\mathit{\mu}$ | z | $\mathbf{\Delta}\mathit{\mu}$ | z | $\mathbf{\Delta}\mathit{\mu}$ | z | |

$\mathit{w}$ | |||||||||

1 | 0.004340 | 3.110 | 0.000241 | 0.051 | 1.053 | 0.079 | 0.833 | 0.167 | |

10 | 0.001349 | 1.364 | 0.000082 | 0.076 | 0.937 | 0.763 | 0.583 | 0.833 | |

100 | 0.000392 | 0.408 | 0.000032 | 0.033 | 0.932 | 0.935 | 1.083 | 1.500 | |

1000 | 0.000106 | 0.111 | 0.000011 | 0.011 | 0.624 | 0.669 | 0.583 | 0.917 |

**Table 12.**Classification quality indicators for the fault detection with both z-score and $\Delta \mu $ for a fault simulation setup with $N=2400$ and ≈80% fault cases under default measurement uncertainty. The classification is evaluated under different filter sizes w and underlying threshold level $\lambda $. Please refer to Table 6 for the definition of the indicators. The graphical illustration visualizes the values for $\lambda =3$, where the corresponding window is marked by colour.

TPR | FPR | FNR | Youden | |||||||

Evaluation | z | $\mathbf{\Delta}\mathit{\mu}$ | z | $\mathbf{\Delta}\mathit{\mu}$ | z | $\mathbf{\Delta}\mathit{\mu}$ | z | $\mathbf{\Delta}\mathit{\mu}$ | ||

$\mathit{w}$ | $\mathit{\lambda}$ | |||||||||

1 | 1 | 0.208 | 0.336 | 0.491 | 0.391 | 0.792 | 0.664 | −0.283 | −0.055 | |

2 | 0.136 | 0.268 | 0.123 | 0.123 | 0.864 | 0.732 | 0.013 | 0.144 | ||

3 | 0.079 | 0.227 | 0.006 | 0.033 | 0.921 | 0.773 | 0.073 | 0.194 | ||

10 | 1 | 0.774 | 0.820 | 0.354 | 0.362 | 0.226 | 0.180 | 0.420 | 0.458 | |

2 | 0.686 | 0.731 | 0.091 | 0.121 | 0.314 | 0.269 | 0.596 | 0.610 | ||

3 | 0.617 | 0.665 | 0.022 | 0.024 | 0.383 | 0.335 | 0.595 | 0.642 | ||

100 | 1 | 0.966 | 0.966 | 0.353 | 0.337 | 0.034 | 0.034 | 0.613 | 0.630 | |

2 | 0.962 | 0.963 | 0.118 | 0.108 | 0.038 | 0.037 | 0.845 | 0.855 | ||

3 | 0.955 | 0.958 | 0.035 | 0.029 | 0.045 | 0.042 | 0.920 | 0.929 | ||

1000 | 1 | 0.941 | 0.946 | 0.379 | 0.391 | 0.059 | 0.054 | 0.562 | 0.555 | |

2 | 0.931 | 0.935 | 0.104 | 0.100 | 0.069 | 0.065 | 0.826 | 0.835 | ||

3 | 0.919 | 0.924 | 0.014 | 0.014 | 0.081 | 0.076 | 0.905 | 0.910 |

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## Share and Cite

**MDPI and ACS Style**

Klink, J.; Grabow, J.; Orazov, N.; Benger, R.; Hauer, I.; Beck, H.-P.
Systematic Approach for the Test Data Generation and Validation of ISC/ESC Detection Methods. *Batteries* **2023**, *9*, 339.
https://doi.org/10.3390/batteries9070339

**AMA Style**

Klink J, Grabow J, Orazov N, Benger R, Hauer I, Beck H-P.
Systematic Approach for the Test Data Generation and Validation of ISC/ESC Detection Methods. *Batteries*. 2023; 9(7):339.
https://doi.org/10.3390/batteries9070339

**Chicago/Turabian Style**

Klink, Jacob, Jens Grabow, Nury Orazov, Ralf Benger, Ines Hauer, and Hans-Peter Beck.
2023. "Systematic Approach for the Test Data Generation and Validation of ISC/ESC Detection Methods" *Batteries* 9, no. 7: 339.
https://doi.org/10.3390/batteries9070339